Theorem bnj1071 35335

Description: Technical lemma for bnj69 35368. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1071.7	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj1071	⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Proof of Theorem bnj1071

Step	Hyp	Ref	Expression
1		bnj1071.7	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj923 35127	. 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
3		nnord 7873	. 2 ⊢ (𝑛 ∈ ω → Ord 𝑛)
4		ordfr 6379	. 2 ⊢ (Ord 𝑛 → E Fr 𝑛)
5	2, 3, 4	3syl 19	1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2150   ∖ cdif 3910  ∅c0 4294  {csn 4594   E cep 5564   Fr wfr 5615  Ord word 6363  ωcom 7865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rab 3424  df-v 3464  df-dif 3916  df-ss 3930  df-uni 4878  df-tr 5224  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-ord 6367  df-on 6368  df-om 7866

This theorem is referenced by:  bnj1030  35345  bnj1133  35347